What's the "best" and easiest way to remove the current left-overs of the seasoning and start all over again? Anything I should keep in mind? Is it better to use a gas stove rather than electrical one when re-seasoning it or does it not matter?

Carbon steel pan seasoning isn't like cast iron seasoning, the metal is less porous & doesn't absord the seasoning the same way. You don't need to strip it. Just wash the pan and reseason. Frankly, I just heat up oil in mine on the stovetop, wipe it out with a paper towel, and let it cool. You can do that every time you wash it.

I prefer using salt and/or a stainless steel scrubber for cleaning all of my CS pans and woks. The last step is to deposit 2 to 3 drops of oil and some salt and rub it with a paper towel and finally rinse and dry.